Finite Block Length Rate-Distortion Theory for the Bernoulli Source with Hamming Distortion: A Tutorial

Abstract

Lossy data compression lies at the heart of modern communication and storage systems. Shannon's rate-distortion theory provides the fundamental limit on how much a source can be compressed at a given fidelity, but it assumes infinitely long block lengths that are never realized in practice. We present a self-contained tutorial on rate-distortion theory for the simplest non-trivial source: a Bernoulli$(p)$ sequence with Hamming distortion. We derive the classical rate-distortion function $RD = Hp - HD$ from first principles, illustrate its computation via the Blahut-Arimoto algorithm, and then develop the finite block length refinements that characterize how the minimum achievable rate approaches the Shannon limit as the block length $n$ grows. The central quantity in this refinement is the \emph{rate-distortion dispersion} $V(D)$, which governs the $O(1/\sqrt{n})$ penalty for operating at finite block lengths. We accompany all theoretical developments with numerical examples and figures generated by accompanying Python scripts.